Chapter 12 Modern Experiments on Atom-Surface Casimir...

Chapter 12Modern Experiments on Atom-SurfaceCasimir Physics

Maarten DeKieviet, Ulrich D. Jentschura and Grzegorz Łach

Abstract In this chapter we review past and current experimental approaches tomeasuring the long-range interaction between atoms and surfaces, the so-calledCasimir-Polder force. These experiments demonstrate the importance of goingbeyond the perfect conductor approximation and stipulate the relevance of theDzyaloshinskii-Lifshitz-Pitaevskii theory. We discuss recent generalizations ofthat theory, that include higher multipole polarizabilities, and present a list ofadditional effects, that may become important in future Casimir-Polder experi-ments. Among the latter, we see great potential for spectroscopic techniques, atominterferometry, and the manipulation of ultra-cold quantum matter (e.g. BEC) nearsurfaces. We address approaches based on quantum reflection and discuss theatomic beam spin-echo experiment as a particular example. Finally, some of theadvantages of Casimir-Polder techniques in comparison to Casimir force mea-surements between macroscopic bodies are presented.

12.1 Introduction

In this chapter we will be dealing with experimental aspects of the Casimir-Polderforce. (See Chap. 11 by Intravaia et al. in this volume for additional discussionsabout the theoretical aspects of the Casimir-Polder force.) Although no strictcriterion can be implied to distinguish between the Casimir and the Casimir-Polder

M. DeKieviet (&) � G. ŁachPhysikalisches Institut der Universität Heidelberg, Philosophenweg 12,69120, Heidelberg, Germanye-mail: [email protected]

U. D. JentschuraDepartment of Physics, Missouri University of Science and Technology,Rolla, MO 65409-0640, USA

D. Dalvit et al. (eds.), Casimir Physics, Lecture Notes in Physics 834,DOI: 10.1007/978-3-642-20288-9_12, � Springer-Verlag Berlin Heidelberg 2011

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effect, it is common use to describe the interaction between two macroscopic,polarizable (neutral and nonmagnetic) bodies due to the exchange of virtualphotons as Casimir interactions. In contrast, Casimir-Polder effects should involveat least one microscopic, polarizable body, typically an atom or molecule.Although the transition from Casimir to Casimir-Polder physics is a continuousone,1 the distinction goes back to the seminal 1948 paper by Casimir and Polder[1]. Herein, the authors discuss the influence of retardation on the London-van derWaals forces. In particular, they derive the long-distance behavior of the quantumelectrodynamic interaction between an atom and a perfectly conducting surfaceand that between two atoms. The works by Casimir and Casimir and Polderaddress the fact that the mutually induced polarization between two neighboringobjects may be delayed as a consequence of the finiteness of the velocity of light.The Casimir and the Casimir-Polder forces could thus semiclassically be termedlong-range retarded dispersion van der Waals forces. Indisputably the two 1948papers by Casimir and Casimir and Polder mark the beginning of a whole newbranch of research addressing fundamental questions about quantum field theory ingeneral and the structure of the vacuum in particular. The concepts developed in1948 are now being used in order to describe a rich field of physics, and have beensupplemented by a variety of methods to address also practically important areasof applications. Especially with the rise of nanotechnology, there is a growing needfor a quantitative understanding of this interaction and experimental tests areindispensable.

12.2 The History of Casimir-Polder Experiments

Before we turn to modern aspects of Casimir-Polder physics it is useful to reviewsome of the developments that led us here. The correct explanation for the non-retarded dispersive van der Waals interaction between two neutral, but polarizablebodies was possible only after quantum mechanics was properly established. Usinga perturbative approach, London showed in 1930 [2] for the first time that theabove mentioned interaction energy is approximately given by

VLondonðzÞ � �3�hx0aA

1 aB1

26p2e20z6

ð12:1Þ

where aA1 and aB

1 are the static dipole polarizabilities of atoms A and B, respec-tively. x0 is the dominant electronic transition frequency and z is the distancebetween the objects. Experiments on colloidal suspensions in the 1940s by Verweyand Overbeek [3] showed that London’s interaction was not correct for largedistances. Motivated by this disagreement, Casimir and Polder were the first in

1 It is in fact a useful and instructive exercise to derive the latter from the former by simpledilution.

394 M. DeKieviet et al.

1948 to consider retardation effects on the van der Waals forces. They showed thatin the retarded regime the van der Waals interaction potential between twoidentical atoms is given by

VretardðzÞ ¼ �23�hcaA

1 aB1

26p3e20z7

: ð12:2Þ

The reason is that, at larger separations, the time needed to exchange infor-mation on the momentary dipolar states between the two objects may becomecomparable to or larger than the typical oscillation period of the fluctuating dipole.The length scale at which this happens can be expressed in terms of a reducedwavelength �k of the lowest allowed atomic transition that relates to the transitionfrequency x0 (or wavelength k) and the speed of light c

z� c

x0¼ k

2p¼ �k: ð12:3Þ

The onset of retardation is thus a property of the system. In their paper, Casimirand Polder also showed that the retarded van der Waals interaction potentialbetween an atom and a perfectly conducting wall falls at large distances as 1=z4, incontrast to the result obtained in the short-distance regime (non-retarded regime),where it is proportional to 1=z3. They derived a complete interaction potentialvalid also for intermediate distances. By the use of the dynamic polarizability ofthe atom aðixÞ the Casimir and Polder result can be rewritten as

V1ðzÞ ¼ � �h

ð4pÞ2e0z3

Z1

0

dxa1ðixÞ�

1þ 2xz

cþ 2�xz

c

�2�

e�2xz=c: ð12:4Þ

Its short-range limit, equivalent to the nonrelativistic approximation (c!1)reproduces the van der Waals result for the atom-surface interaction energy

V1ðzÞ ¼z!0� �h

ð4pÞ2e0z3

Z1

0

dxa1ðixÞ: ð12:5Þ

The long-distance limit for the perfect conductor case is especially importantand has become the signature for the Casimir-Polder force

V1ðzÞ ¼z!1� 3�hcað0Þ25p2e0z4

: ð12:6Þ

In the 1960s, Lifshitz and collaborators developed a general theory of van derWaals forces [4] and extended the result of Casimir and Polder to arbitrary solids.(See also the Chap. 2 by Pitaevskii in this volume for additional discussions aboutthe Lifshitz theory.) Their continuum theory is valid for both dielectrics andsemiconductors, and for conductors, as long as their electromagnetic propertiescan be described by a local eðxÞ. The result for a perfect conductor is recovered by

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taking the limit of infinite permittivity: eðxÞ ! 1. Lifshitz [4] computed theinteraction energy between an atom and a realistic material, described by itsfrequency dependent permittivity evaluated for imaginary frequencies eðixÞ. Hisresult gives the dominant contribution for large atom-surface distances, and reads

VðzÞ ¼ � 2�h

ð4pÞ2e0c3

Z1

0

dxx3a1ðixÞZ1

1

dne�2nxz=cHðn; eðixÞÞ; ð12:7Þ

where the Hðn; eÞ function is defined as

Hðn; eÞ ¼ ð1� 2n2Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2 þ e� 1

p� enffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

n2 þ e� 1p

þ enþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2 þ e� 1

p� nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

n2 þ e� 1p

þ n: ð12:8Þ

The Lifshitz formula (12.7) reproduces the Casimir and Polder result (12.4) forperfect conductors. Both in the short- and the long-range limits, the Lifshitz for-mula simplifies considerably. For z! 0 the interaction potential behaves as

VðzÞ �z!0� �h

ð4pÞ2e0z3

Z1

0

dxa1ðixÞeðixÞ � 1eðixÞ þ 1

� �C3

z3: ð12:9Þ

In the long-distance limit the interaction potential for a generic eðxÞ is

VðzÞ �z!1� 3�hca1ð0Þ2ð4pÞ2e0z4

eð0Þ � 1eð0Þ þ 1

� �C4

z4: ð12:10Þ

Under some special conditions the Casimir and Polder force can also beinferred (both theoretically and experimentally) from the interaction energy of twomacroscopic bodies, if we consider one of the media sufficiently dilute [5]. Theseconditions were fulfilled in the experiments by Sabisky and Anderson [6]. In thissense this 1972 experiment can be viewed as the first experimental verification ofthe Casimir-Polder force. In their beautiful cryogenic experiments Subisky andAnderson measured the thickness of helium films adsorbed on cleaved SrF2 sur-faces in thermal equilibrium at 1.4 K as a function of hyperpressure. The authorsget exceptionally good agreement between the film thickness measured and theone calculated using the Lifshitz formula. The agreement is even more remarkablesince the method strongly depends on the assumption that the non-additivity of thevan der Waals forces does not play a major role. In this case that assumption worksso well because of the extraordinary weakness of the interaction between heliumatoms. The authors note that the best results were obtained on atomically flatregions of the substrate and that the influence of surface roughness in theseexperiments was certainly significant.

Only two years later Shih and Parsegian [7] published an atomic beamdeflection experiment to measure van der Waals forces between heavy alkaliatoms and gold surfaces. With these precision measurements the authors were ablefor the first time to pin down the distance dependency of the van der Waals force in


the non-retarded regime to be proportional to 1=z3: The authors include veryaccurate calculations of the interaction potential performed using the Lifshitzformula. They used ab-initio computed (on the Hartree-Fock level) atomic po-larizabilities and included the finite conductivity of the gold substrate. Theresulting strength coefficient C3 of the van der Waals interaction is not a constantbut varies with separation (the retardation effect). Unfortunately, the measure-ments (see Fig. 12.1) were not precise enough in the region of interest to revealthese retardation effects as a deviation from the 1=z3 behavior. It is worth notingthat although these calculations were the best available at the time the theoreticalvalues they predict are systematically 60% larger than the values for the interac-tion observed in experiments. The authors suggest surface roughness and con-tamination as possible sources for the discrepancy between calculation andexperiment. Still, this seminal paper is the first definite confirmation of the validityof the Dzyaloshinskii-Lifshitz-Pitaevskii [8] formalism for the interaction betweenan atom and a surface, and establishes QED vacuum fluctuations as the commonbasis for van der Waals and Casimir-Polder forces.

It took almost two more decades before a deviation from the 1=z3 behavior for thevan der Waals potential was experimentally resolved. In Ed Hinds’s group [9] abeam of ground state sodium atoms was passed through a micron-sized cavity. Thisgeometry does not exactly correspond to the Casimir-Polder atom single platearrangement, but instead the atoms travel slowly through a cavity made of plates thatinclude a small angle (V-shape). The cavity width can be varied by moving thesource vertically. As a function of plate separation the transmission loss due to the

Fig. 12.1 The Cs beamprofile, normalized to the fullbeam intensity, measured as afunction of the deflectiondistance S, into the geometricshadow of the gold surface.The derived interactionpotential is consistent withthe van der Waals form 1=z3

and has a strength which isbest described within themacroscopic continuumtheory of Lifshitz.(Reproduced with permissionfrom [7].)


long-range interaction with the cavity walls was measured and compared to Monte-Carlo simulations. At large plate distances the purely geometric losses dominate, butbelow a critical width Casimir-Polder losses become significant.2 Hinds et al. couldclearly show a modification of the ground state Lamb shift for the sodium atomswithin the cavity, scaling as 1=z4. Furthermore, the numerical coefficient of the 1=z4

interaction was found to be in agreement with theory. Unfortunately the data do notshow a smooth transition to the expected van der Waals behavior at short distances.The experiment relies heavily on the fact that any atoms not traveling through thecenter of the cavity are accelerated towards one of the two plates, stick there and arethus removed from the beam. This accumulation and its influence on the spectralproperties of the plates were not taken into account.

With the advent of methods for laser cooling and manipulating atoms it becamepossible to use light in order to control ultracold atom-wall collisions. Aspect et al.[10] were the first to use laser cooled and trapped atoms to investigate the Casimir-Polder potential. 87Rb atoms, caught at a temperature of 10 lK; were released froma magneto-optical trap into the gravitational potential. After 15 mm of free fall, theultracold beam of 87Rb atoms impinged at normal incidence on an atomic mirror.This mirror consists of a prism that is irradiated from the back by laser light. Thislaser light forms an evanescent wave extending along the face of the dielectric, andas it is slightly detuned from the atomic resonance frequency of the 87Rb atoms itcan provide a controlled repulsive potential for the atoms. Within this lightpotential the equilibrium between a repulsive light force and the attractive surfaceinteraction is used to establish a potential barrier that can be well controlled bychanging the light field. The reflectivity for incident atoms is measured as afunction of barrier height. Landragin’s data seem to favor the inclusion of retar-dation effects in the atom-wall interaction potential.

In 2001 Shimizu succeeded in reflecting very slow metastable neon atomsspecularly from a solid surface3 [11]. Shimizu prepared an ultracold beam of neonatoms by trapping the metastables in a magneto-optical trap and then releasingthem into the gravitational potential. After tens of centimeters of free fall, themetastables impinged under grazing incidence on the substrate. By varying theangle of incidence he could change the normal incident velocity of the metastableneon atoms between 0 and 35 mm/s. New in his approach was that the quantumreflectivity, that is the reflection from the attractive part of the interaction potentialonly, depends heavily on the perpendicular impinging energy of the particle.Plotting the reflectivity of metastable neon atoms versus the normal incidentvelocity of the beam the author was able to uniquely identify that the attractive

2 It is not entirely clear how much of the nontrivial geometrical effects were taken into accountin the data analysis.3 In early experiments of quantum deflecting H atoms, liquid helium was used as a target.Collectiveness of He atoms within the fluid was ignored, giving information only on the H–Heinteratomic potential. It is not quite clear how many of the exotic properties of this superfluidhave an effect on the interaction potential with this macroscopic quantum object.


interaction was of the Casimir-Polder sort, that is having a 1=z4 dependency.A simple combined van der Waals Casimir-Polder potential was used to fit thedata, but the measurements were not accurate enough to allow for a quantitativecomparison with theory. The surfaces Shimizu used were rough on a scale muchlarger than the de Broglie wavelength of the atoms. This and the fact that the Neatoms were in a metastable state guarantees that none of the atoms is specularlyreflected from the repulsive core potential at short distances.4

In 2003 our group published the first experimental observation of quantumreflection of ground state atoms from a solid surface [12]. Using the extreme sen-sitivity of the earlier developed atomic beam spin echo method [13] we monitoredultracold collisions of 3He atoms from the rough surface of a quartz single crystal.The roughness of the substrate guaranteed that the reflection could only originatefrom the attractive part of the potential. The quantum reflectivity data confirmed thetheory by Friedrich et al. [14] on the asymptotic behavior of the quantum reflectionprobability at incident energies far from the threshold Ei ! 0.5 The data confirmthis high energy asymptotic behavior which is determined by the van der Waalsinteraction and show a gradual transition towards the retarded regime starting asearly as 30 A above the surface. From the data the gas-solid interaction potential isdeduced quantitatively covering both regions. Using a simple approximation for theinhomogeneous attractive branch of the helium-quartz interaction potential

VðzÞ ¼ � C4

z3ðzþ �kÞ ð12:11Þ

we find excellent quantitative agreement with the quantum reflection experimentaldata for the potential coefficient C4 ¼ 23:6 eV Å3 and the reduced atomic tran-sition wavelength �k ¼ 93Å. The experiment establishes quantitative evidence forthe 1=z4 Casimir-Polder attraction and the transition towards the 1=z3 van derWaals regime. Surface roughness was taken into account explicitly, based on anex-situ atomic force microscopic measurement. It is worthwhile stressing that boththe atom and the substrate are in their ground state and no lasers are involved. Itthus represents the situation of a single atom interacting with a well defined,extended, dielectric body through the virtual photons of the quantum fluctuatingvacuum exclusively quite accurately.

The experimental data were also investigated with respect to the inhomogeneityof the potential model used in (12.11). The analysis shows that neither of the twohomogeneous parts, the van der Waals and the Casimir-Polder branches, candescribe the data without the inclusion of the other. Even though at the distancesexplored in this experiment the energies are dominated by the van der Waalsinteraction, there is a definite need to include the Casimir-Polder term explicitly.

4 Mind that metastables may exchange real photons with the substrate.5 For a review on WKB waves far from the semiclassical limit see [15] and references therein.


This stipulates that the onset of retardation is not only smooth, but extends to wellwithin the van der Waals regime.

Around the same period, Perreault and Cronin [16] used an atom interferom-eter, in which a well collimated beam of sodium atoms illuminates a silicon nitridegrating with a period of 100 nm. During passage through the grating slots atomsacquire a phase shift due to the van der Waals interaction with the grating walls.As a result the relative intensities of the matter-wave diffraction peaks deviatefrom those expected for a purely absorbing grating. A complex transmissionfunction was developed to explain the observed diffraction envelopes. By fitting amodified Fresnel optical theory to the experimental data the authors obtain a vander Waals coefficient C3 ¼ 2:7� 0:8 meV nm3 for the interaction between atomicsodium and a silicon nitride surface. A few years later, the experiment was refinedby going to a 50 nm wide cavity. The magnitude of the measured phase shift (seeFig. 12.2) caused by atom-surface interaction is in agreement with that predictedby quantum electrodynamics for a non-retarded van der Waals interaction.

Steady progress in laser cooling and quantum optics and the application ofevaporative cooling at the turn of the century led to the first successful creation ofa macroscopic quantum system. Nowadays Bose-Einstein condensates are pro-duced routinely in dozens of labs all over the world. Their behavior and stability inthe vicinity of a solid wall has been investigated for both fundamental and tech-nical reasons. The first such experiment focusing on the Casimir-Polder interactionwas performed in the group of Ketterle [17] using a Bose-Einstein condensate of23Na atoms. Since the BEC is much colder than the thermal ensemble in a MOT

Fig. 12.2 Interference pattern observed for sodium atoms passing the atom interferometerwithout a grating (middle), or when the grating is inserted into path a (upper) or path b (lower).The dashed line illustrates the measured phase shift of 0.3 rad. From the measured induced phaseshift as a function of atomic velocity, the authors obtain a van der Waals coefficientC3 ¼ 3 meV nm3. (Reproduced with permission from [16].)


(magnetic-optical trap), like the one Shimizu used, the authors observed quantumreflectivity for incident angles up to normal. Another major difference is that thesilicon substrate used in this experiment is approached while the atoms are stilltrapped in a weak gravito-magnetic trap. The measured reflection probability as afunction of incident normal velocity was compared with a numerical simulationdone for sodium atoms interacting with a conducting wall. The comparison showsqualitative agreement. Even though the authors simulate the interaction betweenthe sodium atoms and the semiconductive surface through a C4 coefficient whichcorresponds to a conducting surface, the calculated reflection probability is stillsystematically higher than the measured data points. The experiment confirms the1=z4 behavior but the range of velocities the authors could explore is not largeenough to investigate the region closer to the surface where the potential has 1=z3

dependence. The authors refer to earlier work by Cornell et al. [18] (see alsoChap. 11 by Intravaia et al. in this volume for additional discussions) and Vuletic[19], who discuss serious problems that condensate-based experiments near sur-faces may experience due to the pollution of even only a small number of atoms atthe surface. Spurious electric and magnetic fields caused by the adsorbates maylead to large local anomalies in the interaction potential and may severely limitsensitive Casimir-Polder force measurements.6 A few years later, a refined versionof the experiment was published and interpreted as a measurement of thetemperature dependence of the Casimir-Polder force [20]. In the analysis,however, the temperature dependence of the coverage and the IR properties of theadsorbates were not included. In a recent experiment by Zimmermann et al. [21],which is comparable to the experiment in [10], an unresolved inconsistency in thequantitative comparison between Casimir-Polder theory and experiment is stillpresent.

It is fair to say that in present day Casimir-Polder physics the quantitativecomparison between experiments and theory is limited by the lack of detailedknowledge of the exact experimental parameters like surface roughness andcleanliness, identity of the adsorbates and their influence on the overall atom-surface interaction potential. Consequently, modern experiments on atom-surfaceCasimir physics should not only aim for higher precision, but simultaneously for abetter characterization and control of these experimental conditions.

Before discussing the different experimental approaches in this context, it maybe useful to identify the physics that could possibly be revealed in future Casimir-Polder studies. In the following section we summarize some of the current theo-retical issues.

6 In [17] Ketterle makes the following observation: ‘‘Surfaces are traditionally consideredenemies of cold atoms: Laser cooling and atom optics have developed thanks to magnetic andoptical traps that confine atoms with non-material walls in ultrahigh vacuum environmentsdesigned to prevent contact with the surfaces. Paradoxically, it turns out that in the extremequantum limit of nano Kelvin matter waves, a surface at room temperature might become a usefuldevice to manipulate atoms.’’


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12.3 The Atom-Surface Interaction

12.3.1 Practical Application of the Lifshitz Formula

The properties of the solid enter the formula (12.7) in the form of its frequencydependent permittivity. Although it has been demonstrated that e can in some casesbe computed ab-initio [22], the accuracy and reliability of the obtained results areuncertain at present. Possible difficulties include the contribution of vibrationalexcitations within the solid to the atom-surface interaction energy [23]. This termcan be relatively large, like in the case of glass or a�quartz (10%) and has so farbeen neglected in the full ab-initio approach.

Some of the solids for which the contemporary experiments are performed aretransition metals, for which both the relativistic effects and the fact that metals areopen shell systems make accurate theoretical predictions extremely difficult.Another drawback of the presently available theoretical results for solids is theneglect of temperature effects. As a result the theoretically computed permittivityas a function of frequency typically exhibits more structure than the ones derivedfrom experimental optical data. This makes the use of published optical data auseful alternative to theoretical computation of the dielectric properties of solids(e.g. [24] and references therein). The procedure, however, is complicated by thefact that experimental data for the complex permittivity exist only for real fre-quencies and its values along the imaginary axis have to be reconstructed using theKramers-Kroenig relation:

eðixÞ ¼ 1þ 2p

Z1

0

dnn Im eðnÞx2 þ n2 : ð12:12Þ

Apart from the frequency dependent permittivity of the solid the computation ofthe atom-surface interaction energy requires the dynamic dipole (or multipole)polarizability of the atom as an input. In the past a single resonance model for thedynamic polarizability was used:

a1ðixÞ ¼a1ð0Þ

1þ x2=x21

; ð12:13Þ

where x1 is the lowest allowed transition frequency. This simplification may leadto an error in the short range interaction energy as large as 40% (for the case of He)when compared to results that use more accurate polarizabilities. Slightimprovement is obtained by using a few-resonance approximation, in which a fewlowest lying excited states are included in the spectral expansion:

a1ðixÞ ¼X

i

f ð1Þi0

x2i þ x2

; ð12:14Þ


where xi denote the allowed transition frequencies, and f ð1Þi0 (see for example [23])are the corresponding oscillator strengths. Even when all discrete transition fre-quencies are included, this approximate approach still leads to a 20% error (forground state He) as it neglects the significant contribution coming from excitationsto the continuous spectrum. We therefore recommend using ab-initio computeddynamic atomic polarizabilities whenever possible. Dynamic polarizabilities of thenoble gas, alkali and alkaline atoms evaluated for imaginary frequencies haverecently been published [25].

12.3.2 Limitations of the Lifshitz Theory

The Lifshitz formula describes the interaction of an arbitrary atom with any surfaceand is a good starting point for the comparison between atom-surface theory andexperiment. It should be realized, however, that it suffers from a number ofdrawbacks that may become important in future Casimir-Polder experiments. Aspointed out earlier, formula (12.7) becomes invalid for atom-surface distancescomparable to the charge radius of the atom (of the order of Å), due to exchangeeffects. At slightly larger distances, of the order of a few Å (for ground state atoms,but as large as a few nm for metastable helium atoms), the Lifshitz approach, beingbased on second order perturbation theory, breaks down due to higher order effects.At even larger distances (a few nm for He) its validity is reduced further, whenquadrupole and higher multipole polarizabilities of the atom are to be included.These effects can be theoretically computed and are considered in the next sections.These, and other corrections to the Lifshitz formula, that cannot easily be assigned adefinite length scale will be considered in the following sections. Their list includes:

• Corrections from higher orders of perturbation theory• Contributions from higher multipole polarizabilities of the atom• Temperature effects• Relativistic and radiative corrections• Effects of magnetic susceptibilities of both the atom and the solid• Corrections from the nonplanar geometry and imperfections of the surface

From the experimental point of view some of these effects are already in reachwith current methods. Others, like the relativistic and radiative corrections, or theeffects of nonlocal response of the surface, may only become visible in futuregenerations of experiments.

12.3.3 Higher Orders of Perturbation Theory

The applicability of the Lifshitz formula is limited to distances z larger thanffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia1ð0Þ=ð4pe0Þ3

p, at which higher orders of perturbation theory become important.


The next nonzero contribution to the atom-surface interaction energy comes fromthe fourth order in atom-field coupling, i.e. second order in atomic dipole polar-izability. This has been calculated by Marvin and Toigo [26] and reads

V1;1ðzÞ ¼ ��h

29p2e0z6

Z1

0

dxa21ðixÞ

�eðixÞ � 1eðixÞ þ 1

�2

e�2xz=cP1;1ðxz=cÞ; ð12:15Þ

where the polynomial P1;1ðxÞ is defined as

P1;1ðxÞ ¼ 3þ 12xþ 16x2 þ 8x3 þ 4x2: ð12:16Þ

The f1; 1g index indicates that (12.15) depends on the square of the dipolepolarizability of the atom. It is only the first term of the series of correctionsand the ones depending on products of multipole polarizabilities followed byV1;2; V1;3; V2;2; V1;1;1, etc., all of which are at present not calculated.

It is worth noting that together with the above result the same authors present anincorrect formula for the first order term, given here in (12.7). Their leading orderresult [26] including retardation is in disagreement with [4, 23, 27]. Its short-range,nonrelativistic limit is, however, in agreement with other works. The short-rangelimit of (12.15) is

V1;1ðzÞ ¼ ��h

29p2e0z6

Z1

0

dxa21ðixÞ

�eðixÞ � 1eðixÞ þ 1

�2

ð12:17Þ

and is consistent with the nonrelativistic result of McLachlan [28].

12.3.4 Effect of Multipole Polarizabilities

Another group of effects limiting the use of (12.7) are those resulting from thehigher multipole polarizabilities of the atom, like quadrupole and octupole ones.The corrections from quadrupole polarizability of the atom become important

when z is comparable toffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2ð0Þ=a1ð0Þ

p, where a1 and a2 denote the dipole and

quadrupole polarizabilities of the atom.The effects of the quadrupolar polarizability of the atom on interaction potential

V2ðRÞ were first calculated in the nonrelativistic case by Zaremba and Hutson[29]. In general the second order interaction energy can be written as a multipoleexpansion

VðzÞ ¼V1ðzÞ þV2ðzÞ þV3ðzÞ þ � � � ð12:18Þ

where each of the VnðzÞ terms comes from the 2n-pole atomic polarizabilitycoupled to the fluctuating electromagnetic field. The first term V1ðzÞ is the


previously considered Lifshitz contribution, while the next ones constitute cor-rections, which in the limit of small distances behave as C2nþ1=z2nþ1.

For short distances, higher terms in the multipole expansion used to derive (12.7)become important. The generalization of the derivation of the Lifshitz formula toinclude higher multipoles has been presented in [23], and gives contributions to theinteraction energy from an arbitrary 2n-pole polarizability of the atom:

VnðzÞ ¼ ��h

8p2e0c2nþ1

Z1

0

dx x2nþ1anðixÞZ1

1

dne�2nxz=cPnðnÞHðn; eðixÞÞ;

ð12:19Þ

where anðxÞ is the 2n-pole polarizability and PnðnÞ are polynomials which forn ¼ 1; 2; 3; 4 (dipole, quadrupole, octupole and hexadecupole components)explicitly read

P1ðnÞ ¼ 1; ð12:20aÞ

P2ðnÞ ¼16ð2n2 � 1Þ; ð12:20bÞ

P3ðnÞ ¼1

90ð4n4 � 4n2 þ 1Þ; ð12:20cÞ

P4ðnÞ ¼1

90ð8n6 � 12n4 þ 6n2 � 1Þ: ð12:20dÞ

In the limit of small distances or, equivalently, in the nonrelativistic limit(c!1) (12.19) leads to a surprisingly simple result:

VnðzÞ ¼z!0� �h

ð4pÞ2e0z2nþ1

Z1

0

dxanðixÞeðixÞ � 1eðixÞ þ 1

: ð12:21Þ

The n = 2 case reproduces the short-range asymptotic result of Zaremba andHutson [29]. The derivation of the long-distance limit of the multipole contribu-tions lead to

VnðzÞ ¼z!1� �hcanð0Þ

ð4pÞ2e0z2nþ2Dn


; ð12:22Þ

where the Dn constants are: D1 ¼ 3=8; D2 ¼ 25=12; D3 ¼ 301=120; D4 ¼ 1593=560.The quadrupole and octupole dynamic atomic polarizabilities of helium atoms

(in the ground and metastable states) together with accurate (at 10�8) and simpleglobal fits have been published [30]. Simulations based on potentials computedusing these multipole polarizabilities have shown that, in the quantum reflectionexperiments of the type reported in [12], the quadrupolar contribution could beseen at the few percent level.


12.3.5 Effects of Non-zero Temperature

Modification of the atom-surface interaction energy due to non-zero temperaturehas two distinct origins. The permittivity of the solid, entering the formulae(12.7)–(12.19), is itself temperature dependent. More importantly there is also anexplicit temperature dependence resulting from thermal fluctuations of the elec-tromagnetic field, calculated first by Dzyaloshinskii et al. [8]. The complete for-mula in a form closely resembling (12.7) reads [31, 32]

V1ðz; TÞ ¼ �hkBT

4pe0

X1i¼0

ð1� 12d0iÞa1ðixiÞ

Z1

1

dne�2nxi=c Hðn; eðixiÞÞ; ð12:23Þ

where kB is the Boltzmann constant, xi ¼ 2pikBTz=�h are the Matsubarafrequencies and d0i ¼ 1 for i ¼ 0, and is equal to 0 otherwise. The above resultrelies on the assumption that the temperature is small enough to neglect thermalexcitations of the atom. This assumption is true as long as �hx1 kBT , which isfulfilled for most ground state atoms at any reasonable temperature (i.e. below themelting point of the experimentally considered solids), but does not have to be truefor experiments with metastable excited states.

Just as for the T ¼ 0 case, in the short-range limit formula (12.23) can beapproximated by

V1ðz; TÞ ¼z!0�C3ðTÞz3

; ð12:24Þ

where

C3ðTÞ ¼�hkBT

16pe0að0Þ eð0Þ � 1

eð0Þ þ 1þ 2

X1i¼1

aðixiÞeðixiÞ � 1eðixiÞ þ 1

" #: ð12:25Þ

In the low temperature limit the above result approaches the T ¼ 0 one (12.9).It has already been discovered by Lifshitz that, in comparison to the zero-tem-perature case, the long-distance behavior of (12.23) is qualitatively different. Fordistances much greater than the thermal length scale introduced in Matsubarafrequencies

z kT ��hc

kBT; ð12:26Þ

the atom-surface interaction energy behaves as:

VðzÞ �z!1� kBTa1ð0Þ4ð4pÞe0z3


: ð12:27Þ

It is worth noting that the coefficient at the long distance 1=z3 asymptotics given by(12.27) vanishes for T ! 0, but the range of distances where it is valid, given by


(12.26), also widens. The distance scale of the temperature effects kT is usually ordersof magnitude larger than the retardation distance scale �k and for experimentallyconsidered cases the 1=z4 asymptotics given by (12.6) is valid for a wide range ofdistances. At room temperature kT is on the order of 100 lm (infrared wavelength),while k is on the order of 100 nm for most ground state atoms (ultraviolet wave-length). For conductors, due to the singularity of eðxÞ at zero frequency, (12.27)takes a universal form independent of any characteristics of the solid. It remains asubject of current research whether or not this result requires modification. Thispeculiarity is especially pronounced for conductors with a very low concentration ofcarriers, or with slowly moving carriers (for example ions) [33].

12.3.6 Relativistic and Radiative Corrections

The theory of relativistic corrections to the interaction between closed shell atomswas recently developed by Pachucki [34]. Earlier studies of the interplay betweenrelativistic and radiative corrections had been done by Meath and Hirschfelder [35,36]. The general result is that the relativistic corrections of the lowest order (a2)can be grouped into three categories, where the ones dominant at large interatomicdistances (orbit-orbit, or Breit interaction) are already incorporated in the Casimir-Polder formula. Corrections can be classified as relativistic corrections to dynamicatomic polarizabilities, or as coupling between electric polarizability of one atomand the magnetic susceptibility of the other [34]. The relativistic and QED cor-rections to atomic polarizabilities are, at least for the well studied case of He,negligible at the level of precision considered here [37], and so are contributionsfrom atomic magnetic susceptibility. On the other hand the contribution to theCasimir-Polder force resulting from the magnetic susceptibility of the solid canlead to measurable effects in the case of ferromagnetic materials [38].

12.3.7 Effects of Nonplanar Geometry and Nonuniformity

As described in the introductory chapter, many of the original experiments investi-gating the atom-surface interactions have been performed for geometries differentfrom the planar one. The cases of an atom interacting with macroscopic spheres andcylinders (or wires) have been partly solved [39–41]. For more complicated geom-etries corrections can be estimated, at least for dielectric materials, using the pairwisesummation method [5]. When the radius of curvature of the surface is much largerthan the relevant range of atom-surface distances (which is frequently the case) theresulting geometric corrections are negligible. For all other cases, a first rigoroustheoretical study on the inclusion of uniaxial corrugations has been performed for ascalar field by Gies et al. [42]. Unfortunately the extension of these results to elec-tromagnetic fields is nontrivial. An important exception results from the fact that


even if the experiments are meant to be performed on flat surfaces, the nontrivialgeometry enters in the form of corrections induced by corrugations or imperfectionsof the surface. In the limit of atom-surface distance much larger than the surfacecorrugations the roughness of the surface can be taken into account as a modificationof its reflection coefficients [43].

The case when neither of the conditions mentioned above is fulfilled, whichmeans that the curvature radius of the surface is neither very small nor very largein comparison to the atom-surface distance, and the surface is a conductor, is anopen problem.

Modern Casimir-Polder Experiments

Modern experiments on atom-surface Casimir physics should not only be moresensitive and precise as a technique, but also be able to characterize and control thesystem under investigation in a quantitative manner. The first requirement isobvious; the second, however, no less important. Progress in this field will criti-cally depend on the applicability of direct comparisons between theory andexperiment and thus on the accuracy of the system parameters used on both sides.Let us relate this to the different type of modern Casimir-Polder experimentsaccording to the approach they apply.

• Spectroscopic measurements: It is common knowledge in experimental physicsthat the best way for improving precision is to design a signature that can bemeasured as a frequency. In this respect the (quantum) optics experiments ofthe type mentioned above naturally promise a huge potential for Casimir-Polderphysics. The atom-wall interaction leads to a distance-dependent shift of theatomic energy levels which may be detected by resonant spectroscopic tech-niques. An important and necessary requirement for this technique to work isthat the energy shifts involved for the excited atoms behave very differentlyfrom those of the ground state species. The equilibrium between a repulsivelight force and the attractive surface interaction can be used to establish apotential barrier. The barrier height can be controlled by changing the light fieldand its influence is then probed through the reflectivity for incident atoms [10].For systems involving excited [44, 45] or Rydberg atoms [46, 47] additionalresonance effects come into play, which may in fact turn the Casimir-Polderinteraction itself into a repulsion (see for example [48]). Early experiments onthe direct spectroscopic measurement of van der Waals forces have been suc-cessful up to a level of several tens of a percent [44, 46]. To date, it seemsfeasible to explore the progress in spectroscopic precision and to investigate thecomplex QED effects taking place during the atom-wall collisions an order ofmagnitude more accurately.

• Bose-Einstein condensates (BECs): From the manipulation of a BEC in front ofa wall, information on the single atom-surface interaction can be obtained,


since all particles within this macroscopic quantum system are in the samestate. As the trap-minimum is moved closer towards the surface, the Casimir-Polder forces change the curvature of the trapping potential which leads to arelative change of the center of mass oscillation frequency. Cornell’s groupmeasured this frequency shift for BEC atoms inside a chip-based trap [18, 49].Refined measurements by the same group at the 10�4 level were interpreted byAntezza et al. in terms of equilibrium and nonequilibrium thermal correctionsto the Casimir-Polder force [50]. The interaction strength was determined at aten percent level. In a recent proposal it was argued that the precision in thistype of BEC experiments may be improved by an order of magnitude throughthe use of atomic clocks [51].In both types of quantum optics experiments above it remains unclear how thestate of the surface and its deposits can be quantitatively measured and includedin theory. The same argument also holds for the deflection experiments byHinds et al. that rely on the fact that atoms are removed from the beam throughsticking. Absorbed atoms contaminate the substrate and modify the interactionpotential at a level that need not be constant, but may vary with the experi-mental conditions, like time, light intensity and temperature. Even at sub-monolayer coverage, adsorbates (in particular metallic ones) modify thedielectric properties of the system. This is in fact one of the reasons why IRspectroscopy is such a sensitive and well established tool in surface science (seefor example [52]). Benedek et al. [53] recently reviewed spectral consequencesof vibrations of alkali metal overlayers on metal surfaces, that depend verysensitively on atomic mass, adsorption position, coverage and substrate ori-entation. Pucci et al. [54] have recently found a strong enhancement ofvibration signals by coupling an antenna to surface phonon polaritons, using(far) IR surface spectroscopy. In particular the investigation of temperatureeffects in the Casimir-Polder physics may be strongly influenced by suchspectral features. Inversely, one could include (far) IR spectroscopy as a tool toquantify the state of the substrate in future BEC experiments.

• Atom interference: Another very useful step for improving experimental pre-cision is shifting from intensity measurements to phase sensitive quantities. Theexperiments mentioned in Sect. 2 by Aspect [55] and Cronin [16] are examplesalong this line. At the heart of both techniques is an atom interferometer, whichmeasures phase shifts in the de Broglie waves of the atoms. These firstexperiments to detect surface-induced phase shifts were not accurate enough totest the power law of the potential. Recently, however, by using the Toulouseinterferometer significant improvement could be reported [56]. The interactionstrength between an atom and the silicon nitride nano-grating was now deter-mined with a precision of 6%. Since the setup was not equipped with anysurface analytical tools, the authors used a trick to reduce the effect of surfacecontamination discussed above. By taking ratios of interaction strength coef-ficients obtained for different atoms, they compared the van der Waals systemsat a level of better than 3%. It must be realized, however, that taking the ratio


between two C3 values is good only to first order approximation: differentatomic species adsorb differently onto the same surface, with distinguishedtemperature behavior and spectral features. In comparing the two, the dielectricproperty changes of the wall need not be exactly the same. Still, this method is adefinite improvement and could be useful for other techniques as well.

• Quantum reflection: In 2002 Shimizu et al. reported on a giant quantumreflection of neon atoms from a ridged silicon surface [57]. The enhancement ofreflectivity with respect to that from flat silicon surfaces was initially explainedas resulting from the reduced effective matter density in the outermost region ofthe structured sample. At comparable distances, the interaction potential wouldconsequently be weaker for the latter and that would enhance the so called‘‘Bad Land’’ condition for quantum reflectivity at a given value of z. Later itwas established that the giant reflectivity was not so much a consequence of amodification in the Casimir-Polder potential, but rather due to the collisionprocess itself.7,8 The explanation was modified, interpreting the process ofgrazing incidence over the ridged surface in terms of a Fresnel diffraction [58].This explanation, however, ignores the modification of the Casimir-Polderpotential due to the periodic structure. This prompted us to experimentallyinvestigate the underlying question of including geometric effects in this QEDphenomenon more systematically. Extensive studies on the quantum reflectivityof ground state He atoms from nano-structured substrates of different shape andmaterial were performed. The results have not been published yet, but havealready initiated great activity in theory to develop a method for embeddingthese structures in the scattering formalism. In addition, our experiments showthat the non-additivity of the Casimir-Polder potential leads to a strong azi-muthal dependence of the measured reflectivity. The data allow for a phe-nomenological power law description of the interaction, which has a differentexponent along different azimuthal directions.

12.4.1 The Heidelberg Approach

Thermal atom beam scattering was founded in the early 20th century by Stern andEstermann [59, 60] to verify the wave nature of matter. With the advent of highquality supersonic jets, the method of diffracting a beam of atoms from a crys-talline surface developed in the 80’s into a powerful tool in surface science.

7 The need for a more sophisticated explanation was supported by independent observations inKetterle’s laboratory. In experiments at normal incidence dedicated to extremely low densitymaterials quantum reflection was never observed.8 Recently, similar experiments were performed in Gerhard Meijer’s group, scattering He atomsfrom a periodic micro-structure. It could not be substantiated that the process observed is in factquantum reflection.


‘‘Classical’’ thermal atom scattering9 is governed predominantly by the repulsivepart of the full atom-surface interaction potential (see Fig. 12.3). This repulsion isdue to Pauli exclusion during the partial overlap of the electron clouds of the atomand the solid at very short distances. The full interaction includes of course boththe repulsive and attractive branches and shows a well of minimum potentialenergy.

The angular distribution of the specularly reflected atoms reveals informationon the crystallinity and cleanliness of the exposed surface.10 If the de Brogliewavelength of the atoms is on the order of the lattice constant of the crystal, Braggdiffraction may occur. The location of the diffraction peaks gives the periodicity ofthe surface structure, whereas their intensity distribution contains information onthe corrugation height. With these signatures many detailed studies on clean andadsorbate covered substrates were performed. Within the impressive spectrum of

Fig. 12.3 Schematic representation of the full interaction potential between a neutral atom andthe surface of a solid. At large distances, there is an attraction due to the Casimir-Polder and vander Waals effects. This region (marked QR) can be sampled by measuring the quantumreflectivity of an atomic beam impinging at grazing incidence. Close to the surface the interactionis dominated by an exponential repulsion, due to the Pauli blocking. The fraction of atomsscattering ‘classically’ from this wall can be used to extract in-situ information on the quality andthe state of the surface on a sub-atomic level, like periodicity, corrugation, cleanliness anddynamics. Note that the typical energy ranges for these two processes differ by more than sixorders of magnitude

9 The quotation marks merely indicate that scattering from a repulsive wall can of course bedescribed according to the laws of classical physics. It should be clearly distinguished fromquantum reflection, for which there is no classical analogue.10 Note that in the quantum mechanical formulation of atoms scattering from a repulsive wall,there always is a finite probability for the completely elastic channel.


achievements of this technique there are not only the crystallographic studies withsub-Angstrom resolution, but also a whole range of results on two dimensionaldynamics, like surface phonon modes and charge density waves (to get a flavor ofthese accomplishments, see for example [61, 62] and references therein).

Thermal atom scattering is predominantly sensitive to potential featuresbetween the wall and the van der Waals range. Only limited information on thelong distance Casimir-Polder contribution can be extracted from the scatteringpatterns. Physisorption and chemisorption processes, for example, are generallygoverned by the location and shape of the potential well near its energy minimum.The stronger bound states involved have their classical turning points generallynear to the position of its minimum and normally also lie well within the non-retarded regime. They have been discussed to investigate quadrupolar and non-local effects in the physisorption of rare-gas atoms on metal surfaces [63].

Bound state resonance phenomena in atom-surface scattering [61, 62], how-ever, are determined by the upper bound energy levels within the full interactionpotential of 3. The outermost turning points of these higher levels are located atdistances at which retardation effects may start playing a role. Analyzing allavailable data for helium, atomic and molecular hydrogen, however, Vidali et al.64 established a surprising universality, indicating that these bound states lie stillwithin the van der Waals regime. Choosing the mathematical shape of therepulsive wall to be exponential with distance � expð�z=rÞ and fixing the tail tobe � 1=z3 only, Vidali calculated the Bohr-Sommerfeld quantization condition forall bound state levels and plotted these with the experimental data (see Fig. 12.4).He found that independent of their type (metals, semiconductors and insulators) allmeasured systems fall onto the same curve. This impressive result suggests, thatthe adsorption potential energy functional form nearly universally takes the formof an exponential repulsion and a van der Waals attraction.

It remains worthwhile, however, to search for special cases in which theweakest bound state lies but barely underneath the dissociation limit. The outer-most turning points may then be located at distances at which retardation effectsare indeed important.11 For such systems, bound state resonance experimentscould indeed be a valuable tool in modern Casimir-Polder experimental physics,although maybe not a generic one.

About a decade ago, we designed a machine for performing atomic beam spinecho measurements on surfaces. Details on the experimental setup are givenelsewhere [13]. In this apparatus, the nuclear magnetic moments of a flux of 3Heatoms are manipulated so as to obtain detailed information about (changes in)the velocity distribution of the beam before and after scattering from the surface.The 3He beam is very slow (100 m/s \v\ 200 m/s), since the atomic beamsource is cooled down to 1.1–4.2 K. The 3He kinetic energy thus amounts to

11 A famous example from atomic physics: the weak interaction between two neutral He atomsjust happens to accommodate a single bound level at -150 neV only. As a consequence, the bondlength in the He dimer is close to 45 Å!


200–800 leV, corresponding to a de Broglie wavelength kdB of 10–3 Å, or awavevector 2p=kdB of 0.5–1.5 Å�1. The He atoms are exclusively surface sensitiveand have shown to probe corrugations at the surface down to some 0.01 Å [65].The 3He Atomic Beam Spin Echo method has been demonstrated to be a partic-ularly sensitive and precise tool for characterizing the quality, structure anddynamics of clean, as well as adsorbate covered surfaces. As an example wesummarize some of our results obtained with single crystal gold. In elastic 3Hediffraction experiments on the surface of single crystal Au(111), the dimensions ofthe reconstructed unit cell could be extracted to be (p

ffiffiffi3p

), with p ¼ 21:5� 0:5being expressed in the bulk gold nearest neighbor distance a ¼ 2:885 Å. On top ofthis substrate, adsorbate molecules (coronene) were deposited and the structure atmonolayer coverage was determined to be a commensurate (4 4) superstructurewith respect to the unreconstructed gold surface. In addition, we studied diffusionof the diluted coronene molecules at submonolayer coverages. Their 2-D dynamicswas shown to exhibit continuous and non-continuous (jump) 1-D diffusion. Theactivation barrier to this diffusion was inferred from an Arrhenius analysis of itstemperature dependency. We investigated inelastic contributions to the scatteringprocess and confirmed the existence and dispersion of anomalous phonon modesthat are associated with the reconstruction.

Fig. 12.4 Plot from [64]illustrating the experimentalcorrelation between energiesof atom-surface bound statesin units of potential welldepth jenj = �En=D andJðenÞ ¼ ðnþ 1

2Þ�hp=

ðC1=33 D1=6ð2mÞ1=2Þ

(reproduced withpermission). The solid line isthe prediction given by asimple model potentialVðzÞ ¼ 3=ðu� 3Þe�uz=a �1=ðzþ aÞ3 with constantu and a


With the ‘‘classical’’ reflection governed by the repulsive and van der Waalsregime of the potential, we have at our disposal an in-situ analytical tool fromsurface science. On the other hand, we can investigate He atoms being quantumreflected from the retarded regime and study Casimir-Polder physics. The advan-tages of such a powerful combination are summarized in the following Table 12.1.

In order to illustrate that this unique combination of tools is in fact at ourdisposal contemporaneously, we present here experimental data obtained for singlecrystal gold [66]. The measurements were performed much in the same way as ourquantum reflection data described in Sect. 12.2 for the case of single crystal quartzwere obtained [12]. The gold surface, however, is atomically flat12 and also shows3He reflectivity from the repulsive wall (see Fig. 12.3).

Although both the ‘‘classical’’ and the quantum contributions consist of spec-ularly reflected beams, the two can be distinguished through their line shape in theangular distribution. For each incident wave vector a detector angle scan was madeand the two intensities evaluated. In this manner, the dots in upper set for the‘‘classical’’ and the dots in lower set for the quantum reflectivities in Fig. 12.5were obtained. The first curve in upper set is but a guide-to-the-eye for the clas-sical reflection results. The first middle curve in lower set is an exact calculationbased on the simple potential model used in 12.11, using the strength parameter fora perfect conductor. These quantum reflection data were also investigated withrespect to the necessity for inhomogeneity of the model in 12.11. The additionalcurves in lower set in Fig. 12.5 clearly demonstrate that neither a strict van derWaals potential nor a pure Casimir-Polder attraction describe the data accurately.The dashed red line in lower set accounts for the homogeneous Casimir-Poldercontribution (12.6), again using the interaction strength for a perfect conductor.

Table 12.1 Problems limiting the accuracy of classical Casimir force measurements and cor-responding solutions in the 3He Atomic Beam Spin Echo technique

Casimir versus Casimir-Polder

Characterization of the surface quality Surface science (typical \0.01 monolayer)Quantization of surface roughness Atom diffraction (single crystal)Calibration of probe-surface distance Atomic resolution(plate-plate, sphere-plate) (� 0:01 Å)Control over probe Atomic beam(sphere quality, coating, roughness) (� 1019 He atoms s�1 sr�1)Geometry Atom size � relevant length scale(pl-pl: parallellism, sph-pl: proximity

approx.)(1 Å vs. 1 lm)

Spurious electrostatic effects Neutral single atomsImperfect conductor Any system (metal, semi-conductor,

insulator)Resolution Atom interferometry

12 In fact it is the exact same sample that was used in the surface science experiments reviewedearlier in this section.


The corresponding homogeneous van der Waals interaction would give a quantumreflectivity, which is negligibly small. The second middle curve in lower setrepresents the quantum reflectivity based on a van der Waals interaction only,using the more realistic coefficient of C3 ¼ 0:25 eV Å3. From the figure it is clearthat at the distances explored in this experiment the energies are dominated by theCasimir-Polder interaction. Still, there is a definite need to include the van derWaals term explicitly. This demonstrates that with the Heidelberg approach it isnow possible to resolve features in the potential at a few-percent level.

In summary, with the renaissance in atomic physics a great variety of newexperimental techniques have been steadily improving the accuracy in Casimir-Polder measurements to well below the ten percent level. Current trends and pro-posals promise to be pushing this in the near future to below the one percent barrier.At that level of precision new contributions, like those explicitly dependent ontemperature and higher order effects (additional multipoles, etc), are within reach.This will be important for our understanding of vacuum fluctuations. Tremendoustheoretical progress has been made to include non-trivial boundary conditions totheir spectrum. The feasibility of a direct comparison with measured data requires aquantitative experimental characterization of these conditions as well.

Acknowledgements G.L. would like to acknowledge his support from the Deutsche Fors-chungsgemeinschaft (DFG, contract Je285/5–1). U.J. acknowledges support from the NationalScience Foundation (Grant PHY–8555454) and from the National Institute of Standards andTechnology (precision measurement grant).

Fig. 12.5 Plot of the reflectivity of 3He atoms from the single crystal surface of Au(111). Theupper set of data points (including their statistical errors) represents ‘‘classical’’ reflection fromthe repulsive wall (see text). The lower set of data points and error bars corresponds to thesimultaneously measured quantum reflectivity from the attractive tail of the interaction potential.Results for the corresponding simulation are shown in the lower three solid curves: the top one isbased on the full potential derived from (12.19); the two lower curves are based on thehomogeneous potentials for a perfect conductor, the Casimir-Polder result (12.6) (middle) andvan der Waals result (12.5) (bottom), respectively. Finally, the bottom curve represents thehomogeneous van der Waals result (12.5) for a perfect conductor. (Further details will bepresented in [66].)


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